Question

write the function below in the form ( y = f(u) ) and ( u = g(x) ), then find ( \frac{dy}{dx} ) as a function of ( x ).
( y=(2 x+13)^{8} )
write ( y=(2 x+13)^{8} ) in the form ( y = f(u) ) and ( u = g(x) ). choose the correct functions ( f(u) ) and ( g(x) ) below.
a. ( f(u)=2 u^{8} )( g(x)=x + 13 )
b. ( f(u)=u^{8} )( g(x)=2 x + 13 )
c. ( f(u)=(2 u + 13)^{8} )( g(x)=2 x )
d. ( f(u)=2 u + 13 )( g(x)=x^{8} )

Explanation:

Step1: Analyze the composition of the function

We want to write \(y=(2x + 13)^8\) as \(y = f(u)\) and \(u=g(x)\). If we let \(u = 2x+13\) (so \(g(x)=2x + 13\)) and \(y=f(u)=u^8\).

Step2: Use the chain - rule

The chain - rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
First, find \(\frac{dy}{du}\):
If \(y = f(u)=u^8\), then \(\frac{dy}{du}=8u^{7}\) (using the power rule \(\frac{d}{du}(u^n)=nu^{n - 1}\)).
Second, find \(\frac{du}{dx}\):
If \(u = g(x)=2x + 13\), then \(\frac{du}{dx}=2\) (using the sum rule \(\frac{d}{dx}(ax + b)=a\) where \(a = 2\) and \(b = 13\)).

Step3: Calculate \(\frac{dy}{dx}\)

Substitute \(u = 2x+13\) into \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
\(\frac{dy}{dx}=8u^{7}\cdot2\)
\(=16(2x + 13)^{7}\)

Answer:

B. \(f(u)=u^{8}\), \(g(x)=2x + 13\) and \(\frac{dy}{dx}=16(2x + 13)^{7}\)